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Theorem ruv 4295
Description: The Russell class is equal to the universe  _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
Assertion
Ref Expression
ruv  |-  { x  |  x  e/  x }  =  _V

Proof of Theorem ruv
StepHypRef Expression
1 df-v 2604 . 2  |-  _V  =  { x  |  x  =  x }
2 equid 1630 . . . 4  |-  x  =  x
3 elirrv 4293 . . . . 5  |-  -.  x  e.  x
43nelir 2343 . . . 4  |-  x  e/  x
52, 42th 172 . . 3  |-  ( x  =  x  <->  x  e/  x )
65abbii 2195 . 2  |-  { x  |  x  =  x }  =  { x  |  x  e/  x }
71, 6eqtr2i 2103 1  |-  { x  |  x  e/  x }  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1285   {cab 2068    e/ wnel 2340   _Vcvv 2602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4282
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-v 2604  df-dif 2976  df-sn 3406
This theorem is referenced by:  ruALT  4296
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